The invention relates to image registration and more particularly to dynamic image registration based on optimal estimation and maximum likelihood methods.
As background, signal detection theory will first be considered as it is later shown to pertain to the image fusion, registration problem when relative motion exists between the object of interest and the different image capturing apparatus, FIG. 7 portrays the classical problem of signal detection theory. The signal to be identified in the top plot 700 is zero in the low state but may go into a high state for some interval of time 703. The noise in the second plot 701 is zero mean with constant variance. In the bottom plot 702, the sum of the top two prior figures is rendered. The goal is to measure the bottom plot (signal+noise) and to discern when the signal may be in the high state.
Referring to FIG. 6, the zero mean noise is the probability density plot to the left 601. The signal+noise density is the plot translated to the right 602. The likelihood ratio is obtained by drawing a vertical line and taking the ratio of the value of the signal+noise density to the noise, alone density. If this ratio is greater than one, the choice would be made to select the signal being present (H1 hypothesis) over the choice of noise alone (H0 hypothesis). It can be shown that the maximum likelihood decision-making process involves several types of optimality, e.g. if α is fixed in FIG. 6, the minimum β will result if a maximum likelihood ratio is utilized. In the study of optimal estimation theory, the terms “orthogonal projection” can be considered synonymous with the term “optimal estimation.” This is illustrated in FIG. 8 where the shortest distance 802 from a point A 800 to a line 803 is given by the orthogonal projection (line of length d). This is the distance of the point to the orthogonal projection onto the line. The distance d 802 is a minimum and thus is optimal (in a least squares sense) as judged against any other line, such as the dashed line 801 for comparison. In optimal estimation theory, the error between the estimator and the data is denoted via d. The principal of orthogonal projection is very similar to FIG. 8. The procedure of synthesizing an estimator so that the error vector is minimal (analogous to the smallest distance d in FIG. 8) being least squares optimal (lowest error) and is a desired design procedure. FIG. 8 illustrates why this is so for the case of the Euclidean plane. In general, the orthogonal projection method makes the estimation error (estimator−data) perpendicular to the data.
To generalize the optimal estimation problem introduced, an inner product relationship with vectors, and matrices may be developed. The inner (dot) product of two vectors is defined for vectors a, 900, and b, 901, as follows:<a,b>=a·b=|a||b| cos(θ)  Eq 1 (6)where θ is the angle between the two vectors a and b. This dot product concept is depicted in FIG. 9. The dot product can be viewed as the projection of one vector onto a given vector.
The inner product in FIG. 9 shows the optimal projection of the vector a, 900, onto another vector b. If the dot product (the projection onto) is zero with the length of one point, we say that the vector a is orthogonal to the vector b. If this is true, then we have the optimal projection (shortest error distance), thus the optimal solution. This can be generalized to matrices. Let A1 and A2 be two matrices. The inner product definition for vectors can now be generalized to matrices as follows:<A1,A2>=A1·A2=trace (A1TA2)=trace (A1A2T)  Eq 2where trace is the sum of the diagonal elements and the superscript T indicates matrix transpose. Since for two vectors a and b which are orthogonal, their dot product is zero:<a,b>=a·b=0  Eq 3
To extend this to matrices, define A1 as being orthogonal to A2 if and only if:<A1,A2>=A1·A2=0  Eq 4
Thus in this application, the goal of the optimal estimation problem is to minimize an estimation error defined by the dot product of two matrices as follows:Minimize e(t)=estimation error=<A1,A2>=A1·A2  Eq 5
Applying this to the present invention, the matrix A1 will be a difference matrix and may act like a noise source. The matrix A2 may be a signal+noise difference matrix. The dot product will examine any correlation between these possible matrices. These points are discussed in the next section.
It is important to introduce two possible measures to evaluate the efficacy of the algorithms and techniques developed in this patent application in terms of their relative measure of the orthogonal projection of matrices. Denoting the Frobenious norm of a matrix A (A is of size n rows by m columns) as follows:
                                                      A                                F          2                =                              ∑                          i              ,              j                                      n              ,              m                                ⁢                                          ⁢                                    (                              a                ij                            )                        2                                              Eq        ⁢                                  ⁢        6            
The first measure is related to the cosine of the angle between the two matrices A1 and A2:
                              Measure_          ⁢          1                =                              Cosine            ⁢                                                  ⁢                          (                              Angle                ⁢                                                                  ⁢                between                ⁢                                                                  ⁢                                  A                  1                                ⁢                                                                  ⁢                and                ⁢                                                                  ⁢                                  A                  2                                            )                                =                                    trace              ⁡                              (                                                      A                    1                    T                                    ⁢                                      A                    2                                                  )                                                                                                                            A                    1                                                                    F                            ⁢                                                                                      A                    2                                                                    F                                                                        Eq        ⁢                                  ⁢        7            
From the equation above it is shown that 0≦Measure—1≦1 and Measure—1 acts like a cosine of an angle between the two matrices A1 and A2. If Measure—1=0, then the matrices A1 and A2 are orthogonal. Their dot product would then be zero and this would result in an optimal design. A second measure is also useful for determining the degree of orthogonally between matrices.
                              Measure_          ⁢          2                =                              trace            ⁡                          (                                                A                  1                  T                                ⁢                                  A                  2                                            )                                                          1              2                        ⁢                          (                                                                                                              A                      1                                                                            F                                ⁢                                                                                                A                      2                                                                            F                                            )                                                          Eq        ⁢                                  ⁢        8            
Again, it can be shown that 0≦Measure—2<1 and if Measure—2≈0, then the matrices A1 and A2 are orthogonal. Appendix B describes these points in further detail.
The application of this theory is now presented as it pertains to the image fusion, registration problem when relative motion exists between the object of interest and the different image capturing apparatus, according to the arrangement of the present invention.